An essential part of any quantum error correction procedure is the decoder, which specifies the recovery operation that has to be applied for given syndrome data.
The two-layer structure of the error correction described above requires two stages of decoding.
The first of these stages translates continuous GKP-stabilizer syndrome data into the operations required to return to the GKP code subspace housed in each mode, up to qubit-level errors.
It can also provide some detailed information about the relative likelihood of different discrete qubit-level errors [1]}.
The second stage maps syndrome data obtained from measuring the higher-level qubit-code stabilizers to a qubit-level recovery operation.
To avoid confusion, we refer to the former as the inner decoder, and the latter as the outer decoder.
Decoders that are tailored for our architecture are described in more detail in subsec:translator,sec:outer, respectively.
